Finite presentation of alternating groups

Graham Higman posed the question: How small can the integersp andq be made, while maintaining the property that all but finitly many alternating and symmetric groups are factor groups of Δ(2,p,q)=<x,y:x ²=y ^p =3 (xy) ^q =1>? He proved that for a sufficiently largen, the alternating group is a homomorphic image of the triangle group Δ(2,p,q) wherep=3 andq=7. Later, his result was generalized by proving the result forp=3 andq≥7. Choosingp=4 andq≥17 in this paper we have answered the “Hiqman Question”.